{* Closures *}

rc(R)  = refl(R).
sc(R)  = R | R^.
tc(R)  = trans(R).              
rtc(R) = refl(trans(R)).
aec(R) = rtc(sc(R)).

{* Irreflexive part and related problems *}

ipa(R)   = R & -I(R).
hasse(R) = ipa(R) & -(ipa(R)*ipa(R)).
tik(R)   = R & -(R*trans(R)).

{* Base operations on graphs *}

vertex(R)     = point(R).
arc(R)        = atom(R).
edge(R)       = atom(R) | atom(R)^.
roots(R)      = -dom(-rtc(R)).
sources(R)    = sinks(R^).
sinks(R)      = -dom(R).
isolated(R)   = sources(R) & sinks(R).
pendant(R)    = dom(up(R)).
rigid(R)      = dom(mp(R)).
preds(R,v)    = R * v.
succs(R,v)    = R^ * v.
reach(R,v)    = rtc(R)^ * v.
direct(R)     = ipa(R & - rtc(succ(dom(R)))).
undirect(R)   = sc(R) & -I(R).
subgraph(R,v) = R & v * v^.
delete(R,v)   = inj(-v) * R * inj(-v)^.
B(R,S)        = conc(O(R*S)+S,R+O(S*R)).
union(R,S)    = conc(R,On1(R)*O1n(S)) + conc(On1(S)*O1n(R),S).

{* Base operations on relations *}

up(R)         = R & -(R * -I(R^*R)).
mp(R)         = R & R * -I(R^*R).
ran(R)        = dom(R^).
symmdiff(R,S) = (R & -S) | (S & -R).

{* Base operations on order relations *}

max(R,v) = min(R^,v).
min(R,v) = v & ((R & -I(R)) \ -v).
ma(R,v)  = mi(R^,v).
mi(R,v)  = R / v^.
sup(R,v) = inf(R^,v).
inf(R,v) = ge(R,mi(R,v)).
ge(R,v)  = v & -((-R)^ * v).
le(R,v)  = v & -(-R * v).

{* Relations as sequences of vectors *}

first(R)  = R * init(O1n(R)^).
rest(R)   = R * inj(-init(O1n(R)^))^.
conc(R,Y) = (R^ + Y^)^.
sel1(R)   = first(R).
sel2(R)   = R * next(init(O1n(R)^)).
sel3(R)   = R * next(next(init(O1n(R)^))).
sel4(R)   = R * next(next(next(init(O1n(R)^)))).
sel5(R)   = R * next(next(next(next(init(O1n(R)^))))).
sel6(R)   = R * next(next(next(next(next(init(O1n(R)^)))))).
sel7(R)   = R * next(next(next(next(next(next(init(O1n(R)^))))))).
sel8(R)   = R * next(next(next(next(next(next(next(init(O1n(R)^)))))))).
sel9(R)   = R * next(next(next(next(next(next(next(next(init(O1n(R)^))))))))).

{* Relation-algebraic properties *}

universal(R)    = empty(-R).
vector(R)       = eq(R,dom(R)*L1n(R)).
total(R)        = universal(dom(R)).
mapping(R)      = unival(R) & total(R).
injective(R)    = unival(R^).
surjective(R)   = total(R^).
bijective(R)    = injective(R) & surjective(R).
reflexive(R)    = incl(I(R),R).
irreflexive(R)  = incl(R,-I(R)).
symmetric(R)    = incl(R^,R).
transitive(R)   = incl(R*R,R).
equivalence(R)  = reflexive(R) & symmetric(R) & transitive(R).
antisymm(R)     = incl(R&R^,I(R)).
order(R)        = reflexive(R) & antisymm(R) & transitive(R).
difunctional(R) = incl(R*R^*R,R).

{* Graph-theoretic properties *}

simple(R)      = symmmetric(R) & irreflexive(R).
connect(R)     = incl(L(R),rtc(sc(R))).
strconnect(R)  = incl(L(R),rtc(R)).
acyclic(R)     = incl(tc(R), -I(R)).
dirforest(R)   = incl(R*R^,I(R)) & acyclic(R).
dirtree(R)     = dirforest(R) & connect(R).
nooddcycles(R) = incl(rtc(R*R)*R,-I(R)).

{* Diverse functions *}

l(X) = dom(L(X)).
o(X) = dom(O(X)).
bipartit(X) = incl(rtc(sc(X)*sc(X))*sc(X) & I(X),O(X)).
cycfree(X) = incl(tc(X), -I(X)).
forest(X) = incl(X*X^,I(X)) & cycfree(X).
tree(X) = forest(X) & connect(X).
sconnect(X) = incl(L(X),rtc(X)).
isempty(X) = incl(X,O(X)).

{* Random functions *}
randompoint(v)
DECL P
BEG
  P = randomperm(v)
  RETURN P * point(P^ * v)
END.

randomatom(R)
DECL P1, P2
BEG
  P1 = randomperm(O(R));
  P2 = randomperm((O(R))^)
  RETURN P1 * atom(P1^ * R * P2^) * P2
END.

randomcycle(v)
DECL P, C, E
BEG
  C = succ(v);
  P = randomperm(v);
  E = -dom(C) * init(On1(C))^ | C
  RETURN P * E * P^
END.